perplexus.info :: Just Math : Ratio Resolution II

Ratio Resolution II (Posted on 2012-11-24)

Each of x, y, z, a, b and c is a positive real number that satisfy:

(ay-bx)/c = (cx-az)/b = (bz-cy)/a

Determine with proof, the ratio x:y:z in terms of a, b and c.

Submitted by K Sengupta

Rating: 4.0000 (1 votes)

Solution: (Hide)

Let (ay-bx)/c = (cx-az)/b = (bz-cy)/a = m(say)

Then, we have:

(ay-bx) = cm ----(i)
(cx-az) = bm ----(ii)
(bz-cy) = am ----(iii)

(i)*c +(ii)*b+(iii)*a gives:
m(a²+b²+c²) = 0

Since each of a, b and c is positive and therefore nonzero, it follows that m=0, so that:
(ay-bx, cx-az, bz-cy)=(0, 0, 0), and accordingly:
x/y=a/b, z/x= c/a and, y/z = b/c

Consequently, x:y:z = a:b:c

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
Some vector thoughts (spoiler)	Harry	2012-11-27 14:04:33
re: I think I got it.	K Sengupta	2012-11-25 03:32:28
I think I got it.	Jer	2012-11-25 00:03:57

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